Problem: Simplify; express your answer in exponential form. Assume $z\neq 0, t\neq 0$. $\dfrac{{(z^{-5}t^{4})^{-1}}}{{(z^{4}t^{-5})^{-4}}}$
Explanation: To start, try simplifying the numerator and the denominator independently. In the numerator, we can use the distributive property of exponents. ${(z^{-5}t^{4})^{-1} = (z^{-5})^{-1}(t^{4})^{-1}}$ On the left, we have ${z^{-5}}$ to the exponent ${-1}$ . Now ${-5 \times -1 = 5}$ , so ${(z^{-5})^{-1} = z^{5}}$ Apply the ideas above to simplify the equation. $\dfrac{{(z^{-5}t^{4})^{-1}}}{{(z^{4}t^{-5})^{-4}}} = \dfrac{{z^{5}t^{-4}}}{{z^{-16}t^{20}}}$ Break up the equation by variable and simplify. $\dfrac{{z^{5}t^{-4}}}{{z^{-16}t^{20}}} = \dfrac{{z^{5}}}{{z^{-16}}} \cdot \dfrac{{t^{-4}}}{{t^{20}}} = z^{{5} - {(-16)}} \cdot t^{{-4} - {20}} = z^{21}t^{-24}$